# Payoff of option

Consider the payoff $g(S_T)$ shown the figure: I believe the payoff represented as a linear combination of the payoffs of some options with different strike and same maturity $T$ is $$g(S_T) = (2K - S_T)_{+} - (K - S_T)_{+}$$

I am not exactly sure this is correct, any suggestions is greatly appreciated.

• Just plug $S_T=0$ in your equation to see that it does not give $g(S_T=0)=K$ as it should... How can you not be sure that this is incorrect Morgan? The payoff is equivalent to: long put struck at $2K$ and short put struck at $K$ (put spread). – Quantuple Apr 24 '16 at 14:26
• I know what the strategy is but I guess a classmate of mine mistakenly told me that $g(S_T = 0) = 0$ and the $-K$ is that amount of cash we put up for the position. – Wolfy Apr 24 '16 at 14:46
• If you know then why ask? – Quantuple Apr 24 '16 at 20:14
• Well, I just wanted to check if it was right. My professor tends to make many mistakes in his lecture notes – Wolfy Apr 25 '16 at 0:45

## 1 Answer

For this type of question, you basically need only to write the payoff with certain indicator functions. In particular, for the above payoff, we have that \begin{align*} \textrm{Payoff} &= K\, 1_{S_T \le K} + (2K-S_T)\,1_{K < S_T \le 2K}\\ &=K\, 1_{S_T \le K} + (2K-S_T)\big(1_{S_T \le 2K} - 1_{S_T \le K} \big)\\ &=(2K-S_T)\,1_{S_T \le 2K} - (K-S_T)1_{S_T \le K}\\ &=(2K-S_T)^+ - (K-S_T)^+. \end{align*}